
130 Image Processing for Cinema
In what follows, we approximate the curvature κ(I) and unit normal field
η(I) = (η
1
, η
2
) of the image I using forward-backward differences, so
κ(I(x, y)) ' ∆
x
−
∆
x
+
I(x, y)
|∇I(x, y)|
+ ∆
y
−
∆
y
+
I(x, y)
|∇I(x, y)|
(5.7)
and
~η(I(x, y)) = (η
1
(x, y), η
2
(x, y)) '
∆
x
+
I(x, y)
|∇I(x, y)|
,
∆
y
+
I(x, y)
|∇I(x, y)|
, (5.8)
where
∆
x
±
I(x, y) = ±(I(x ± 1, y) − I(x, y)) , ∆
y
±
I(x, y) = ±(I(x, y ± 1) − I(x, y))
and where the discrete gradient is implied we use forward differences, so
|∇I(x, y)| =
q
(∆
x
+
I(x, y))
2
+ (∆
y
+
I(x, y))
2
+
2
for a small > 0. In this setting, we have the following.
Proposition 1. At locations in the image domain where I = a + ...